The local Euler obstruction function appears in the formulation of the local index theorem, which in the above notations asserts the existence of the following commutative diagram (e.g., see [1]} and the references therein):
\({K_0(D^b_c(X)) [d]_{CC} [r]^{\chi _{st}} & F(X) \\LCZ(T^*X) [r]_T^{\cong } & Z(X) [u]_{Eu}^{\cong }}\) 
